1. A Theory for the dynamical origin of intermittency in kinetic Alfvén wave turbulence
P.W. Terry
K.W. Smith
University of Wisconsin-Madison
Outline
Motivation
Alfvénic turbulence and kinetic Alfvén wave turbulence
Intermittency in decaying KAW turbulence
Theoretical description/dynamical origins
CMSO Workshop on Intermittency in Magnetic Turbulence, June 20-21, 2005

2. Motivation
Pulsar signal broadening is consistent with intermittent turbulence in ISM (Boldyrev et al.)
Pulsar signal dispersed by electron density fluctuations in ISM
Broadened signal width scales as R (R: distance from source)
If pdf of electron density fluctuations is Gaussian, signal width ~ R2
If pdf is Levy distributed (long tail), can recover R4 scaling
Key question: is there dynamical path to intermittent density fluctuations in magnetic turbulence?
How do density fluctuations evolve in magnetic turbulence?
Can density fluctuations become large?
At what scale?
Can they become intermittent?
What is the physics?
Recover scaling of broadened pulse from dynamical intermittency?

3. Study intermittency in kinetic Alfvén wave turbulence model
Motivation
Study intermittency in kinetic Alfvén wave turbulence model
Reduced MHD + ñe
Large scale (k<<ri-1): system evolves as MHD + passive ñe
Small scale (k>>ri-1): flow decoupleskinetic Alfvén waves
Can be modeled by coupled ñe and B
2-field intermittency study (Craddock, et al., 1991)
Decaying turbulence
Resistivity << diffusivity
Observed: coherent current filaments, high kurtosis
Questions
Can density structures form? Effect of high diffusivity?
Steady state?

4. Pursue a physical approach to intermittency
Motivation
Pursue a physical approach to intermittency
Common approaches:
Characterize observations (structure functions, etc.)
Statistical (statistical ansatz or theory for pdf)
Physical questions:
How do coherent structures avoid turbulent mixing?
What differentiates them from surrounding turbulence?
3D NS Vortex tube stretching
MHD Self pinch by Lorentz force (dv/dt~JB)
2D NS No inward flow on stable structures, but shear of
azimuthal flow suppresses mixing
KAW No flow, no Lorentz force, no vortex tube stretch
What is mechanism?

5. How does electron density evolve in magnetic turbulence?
Kinetic Alfvén wave turbulence
How does electron density evolve in magnetic turbulence?
Basic model: RMHD + compressible electron continuity
where
Field
Term (large scale)
Turbulent Alfvén wave
Term (small scale)
Kinetic Alfvén wave
Electron density
Elec. advection ( flux): vn
|| compr. (|| flux):B J
Flow
Lorentz force: B J
Ion advection: vv
Magnetic field
Parallel electric field: B f
Elec. pressure: B  n

6. Kinetic Alfvén wave turbulence
Fluctuations change character across the gyroradius scale (Fernandez et al.)
Large scales (kri << 1): Alfvén waves
Coupling: B and v
Density: advected passively (no reaction back on B or v)
Intermittency: Primary structure is current filament
Ancillary structure in vorticity
Density tracks flow; flow is integral of vorticity
 density not strongly intermittent
Small scales (kri > 0.1 in ISM): Kinetic Alfvén waves
Coupling: B and n
Flow: Dominated by self advection, decouples from B and n
Intermittency: Under study

7. Kinetic Alfvén wave turbulence
Electron density fluctuations increase in kinetic Alfvén regime as n equipartitions with B
Spectral energy transfer:
kri << 1: transfer dominated by v  B
kri  1: transfer dominated by n  B
Low k:
v and B equipartitioned
Density at level dictated by
High k:
n and B equipartitioned, even if no
linear or external drive of density
v and B decouple
Low k - high k crossover at kri < 1

8. Kinetic Alfvén wave (KAW) modeled by two-field system for B and n
Kinetic Alfvén wave turbulence
Kinetic Alfvén wave (KAW) modeled by two-field system for B and n
where
No ion flow; ions are fixed, neutralizing background
Damping is ad hoc in n, allows regimes analogous to low, high magnetic Prandtl number of MDH
Previous study (Craddock et al.) 2D
Decaying turbulence
u>> 
Coherent structures observed in current

9. Intermittent current structures emerged as turbulence decayed
Intermittency
Intermittent current structures emerged as turbulence decayed
Kurtosis of current
Current contours
Cuts across
structure
Current structures are localized, small scale, circular
No intermittency in flux (not localized, kurtosis of 3)
Density intermittency not described, but damping was large

10. Intermittency
Intermittency in KAW turbulence has similarities with decaying 2D Navier Stokes turbulence
Decaying 2D NS turbulence (McWilliams):
Structures emerged from Gaussian start up
Kurtosis in vorticity increased to 30
Stream function stayed Gaussian
Cascade connection:
enstrophy (large kurtosis)small scale
energy (small kurtosis)large scale
Structures:
Gaussian curvature profile CT(r)
(mean sq. shear stress - vorticity sq.)
Core: CT <<1
Edge: CT >>1

11. Consider coherent vortex:
Dynamics
Theory: Strong shear in edge of intense localized vortex disrupts mixing by ambient turbulence (Terry et al.)
Consider coherent vortex:
Stability  vortex is circular  flow is azimuthal
Localized vorticity (zero at some radius)  shear is large at edge
Vortex long lived  it is equilibrium for turbulent eddies
Closure + WKB: In strong vortex shear, turbulence is localized to periphery of vortex
 mixing only in edge of vortex  vortex decays very slowly relative to turbulence
Localization of turbulence at edge of vortex 
vortex + turbulence: CT << core
CT >> edge

12. Similar process operates in KAW turbulence (with certain differences)
Dynamics
Similar process operates in KAW turbulence (with certain differences)
No flow, but localized J of coherent structure creates inhomogen-eous B (VA) that localizes turbulence away from structure
KAW
2D Navier Stokes
Localized coherent structure (origin at center of structure)
Inhomogeneous azimu-thal “flow” of structure
Turbulence
Kinetic Alfven waves
Turbulent eddies
Turbulence source
Agent that localizes turbulence
Inhomogeneity of field Bq
in which KAW propagate
Shear flow Vq of vortex

13. Describe with two time scale analysis
Dynamics
Describe with two time scale analysis
Slow time scale:
Evolution of coherent current filament under mixing produced by inhomogeneous KAW turbulence
Rapid time scale:
Radial structure of KAW turbulence in quasi stationary magnetic field of coherent current filament
Assume coherent structure is azimuthally symmetric, turbulence is not azimuthally symmetric (origin at center of structure)
Look at large scales where NL times exceed dissipation times
Show:
KAW turbulence is localized away from coherent structure
Evolution of coherent structure under mixing by KAW turbulence is slow
Show if there is coherent structure in n, in addition to J

14. Use Fourier-Laplace transform to distinguish two time scales
Dynamics
Use Fourier-Laplace transform to distinguish two time scales
 Turbulence: rapidly evolving
Azimuthal variation
Where
Describe rapid evolution with Fourier-Laplace transform:
Recover slow evolution from average over t:
 Structure: Slowly evolving
Azimuthally symmetric

15. Dynamics
Slow time evolution is governed by turbulent mixing of rapidly evolving KAW fluctuations
Solve for turbulent quantities (in vicinity of structure) as nonlinear
response to structure gradients
where P-1 is a turbulent response

16. Dynamics
Fast time equations: nonlinear Alfvén waves propagating in magnetic field of structure, driven by gradients of structure
Alfvén waves propagate in inhomogeneous medium of structure
When structure has strong variation, Alfvénic turbulence acquires radial envelope with narrow width
Seen from balance

17. Dynamics
As variation in structure field becomes large, turbulence localizes to narrow layer
In circular structure, waves propagate without distortion of phase fronts if B varies linearly with r
Deviation from B~ r describes shear field
that distorts propagation
Expand B:
If shear becomes very large, turbulence must become localized in narrow layer r = (r-r0) :  B

18. Dynamics
Localization strongly diminishes turbulent mixing relative to mixing rate outside structure
 Long-lived structures are those with strong shear in B
Turbulent mixing is weaker because:
1) Turbulence localized in narrow layer to periphery of structure where structure has strongest variation (strongest turbulent source n0/r)
For a given source strength n0/r, turbulence becomes weaker as r becomes narrower
Diminishing of mixing governed by balances ,
 coherent structure form in both j and n

19. Conclusions
Small scale kinetic Alfvén turbulence can be modeled with two-field equation
Applicable for ki > 0.1
Simulations show formation of coherent current filaments in decaying turbulence (large electron diffusivity)
Theory shows:
•Coherent structures form because mixing is suppressed if structure has large magnetic field shear
•Turbulence in structure is kinetic Alfvén wave propagating in inhomogeneous medium
•Turbulence is weak and localized to structure periphery when shear in structure field is large  structures avoid mixing
Suppression of mixing applies both to current and density
KAW turbulence dynamics  non Gaussian density fluctuations